GeneralLinearModelAlgorithm¶
(Source code, png, hires.png, pdf)

class
GeneralLinearModelAlgorithm
(*args)¶ Algorithm for the evaluation of general linear models.
 Available constructors:
GeneralLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis, keepCovariance=True)
GeneralLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basisCollection, keepCovariance=True)
 Parameters
 inputSample, outputSample
Sample
or 2darray The samples and .
 basis
Basis
Functional basis to estimate the trend: .
If , the same basis is used for each marginal output.
 basisCollectioncollection of
Basis
Collection of functional basis: one basis for each marginal output.
An empty collection means that no trend is estimated.
 covarianceModel
CovarianceModel
Covariance model of the Gaussian process. See notes for the details.
 keepCovariancebool, optional
Indicates whether the covariance matrix has to be stored in the result structure GeneralLinearModelResult. Default value is set in resource map key GeneralLinearModelAlgorithmKeepCovariance
 inputSample, outputSample
Notes
We suppose we have a sample where for all , with a given function.
The objective is to build a metamodel , using a general linear model: the sample is considered as the restriction of a Gaussian process on . The Gaussian process is defined by:
where:
with and the trend functions.
is a Gaussian process of dimension with zero mean and covariance function (see
CovarianceModel
for the notations).We note:
The GeneralLinearModelAlgorithm class estimates the coefficients and where is the vector of parameters of the covariance model (a subset of ) that has been declared as active (by default, the full vectors and ).
The estimation is done by maximizing the reduced loglikelihood of the model, see its expression below.
Estimation of the parameters and
We note:
where .
The model likelihood writes:
If is the Cholesky factor of , ie the lower triangular matrix with positive diagonal such that , then:
(1)¶
The maximization of (1) leads to the following optimality condition for :
This expression of as a function of is taken as a general relation between and and is substituted into (1), leading to a reduced loglikelihood function depending solely on .
In the particular case where and is a part of , then a further reduction is possible. In this case, if is the vector in which has been substituted by 1, then:
showing that is a function of only, and the optimality condition for reads:
which leads to a further reduction of the loglikelihood function where both and are replaced by their expression in terms of .
The default optimizer is
TNC
and can be changed thanks to the setOptimizationAlgorithm method. User could also change the default optimization solver by setting the GeneralLinearModelAlgorithmDefaultOptimizationAlgorithm resource map key to one of theNLopt
solver names.It is also possible to proceed as follows:
ask for the reduced loglikelihood function of the GeneralLinearModelAlgorithm thanks to the getObjectiveFunction() method
optimize it with respect to the parameters and using any optimization algorithms (that can take into account some additional constraints if needed)
set the optimal parameter value into the covariance model used in the GeneralLinearModelAlgorithm
tell the algorithm not to optimize the parameter using setOptimizeParameters
 The behaviour of the reduction is controlled by the following keys in
ResourceMap
: ResourceMap.SetAsBool(‘GeneralLinearModelAlgorithmUseAnalyticalAmplitudeEstimate’, True) to use the reduction associated to . It has no effect if or if and is not part of
ResourceMap.SetAsBool(‘GeneralLinearModelAlgorithmUnbiasedVariance’, True) allows to use the unbiased estimate of where is replaced by in the optimality condition for .
With huge samples, the hierarchical matrix implementation could be used if OpenTURNS had been compiled with hmatoss support.
This implementation, which is based on a compressed representation of an approximated covariance matrix (and its Cholesky factor), has a better complexity both in terms of memory requirements and floating point operations. To use it, the GeneralLinearModelAlgorithmLinearAlgebra resource map key should be instancied to HMAT. Default value of the key is LAPACK.
A known centered gaussian observation noise can be taken into account with
setNoise()
:Examples
Create the model and the samples:
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x'], ['x+x * sin(x)']) >>> inputSample = ot.Sample([[1.0], [3.0], [5.0], [6.0], [7.0], [8.0]]) >>> outputSample = f(inputSample)
Create the algorithm:
>>> f1 = ot.SymbolicFunction(['x'], ['sin(x)']) >>> f2 = ot.SymbolicFunction(['x'], ['x']) >>> f3 = ot.SymbolicFunction(['x'], ['cos(x)']) >>> basis = ot.Basis([f1,f2, f3]) >>> covarianceModel = ot.SquaredExponential([1.0]) >>> covarianceModel.setActiveParameter([]) >>> algo = ot.GeneralLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis) >>> algo.run()
Get the resulting meta model:
>>> result = algo.getResult() >>> metamodel = result.getMetaModel()
Methods
BuildDistribution
(inputSample)Recover the distribution, with metamodel performance in mind.
Accessor to the object’s name.
Accessor to the joint probability density function of the physical input vector.
getId
()Accessor to the object’s id.
Accessor to the input sample.
getName
()Accessor to the object’s name.
getNoise
()Observation noise variance accessor.
Accessor to the loglikelihood function that writes as argument of the covariance’s model parameters.
Accessor to solver used to optimize the covariance model parameters.
Optimization bounds accessor.
Accessor to the covariance model parameters optimization flag.
Accessor to the output sample.
Get the results of the metamodel computation.
Accessor to the object’s shadowed id.
Accessor to the object’s visibility state.
hasName
()Test if the object is named.
Test if the object has a distinguishable name.
run
()Compute the response surface.
setDistribution
(distribution)Accessor to the joint probability density function of the physical input vector.
setName
(name)Accessor to the object’s name.
setNoise
(noise)Observation noise variance accessor.
setOptimizationAlgorithm
(solver)Accessor to the solver used to optimize the covariance model parameters.
setOptimizationBounds
(optimizationBounds)Optimization bounds accessor.
setOptimizeParameters
(optimizeParameters)Accessor to the covariance model parameters optimization flag.
setShadowedId
(id)Accessor to the object’s shadowed id.
setVisibility
(visible)Accessor to the object’s visibility state.

__init__
(*args)¶ Initialize self. See help(type(self)) for accurate signature.

static
BuildDistribution
(inputSample)¶ Recover the distribution, with metamodel performance in mind.
For each marginal, find the best 1d continuous parametric model else fallback to the use of a nonparametric one.
The selection is done as follow:
We start with a list of all parametric models (all factories)
For each model, we estimate its parameters if feasible.
We check then if model is valid, ie if its Kolmogorov score exceeds a threshold fixed in the MetaModelAlgorithmPValueThreshold ResourceMap key. Default value is 5%
We sort all valid models and return the one with the optimal criterion.
For the last step, the criterion might be BIC, AIC or AICC. The specification of the criterion is done through the MetaModelAlgorithmModelSelectionCriterion ResourceMap key. Default value is fixed to BIC. Note that if there is no valid candidate, we estimate a nonparametric model (
KernelSmoothing
orHistogram
). The MetaModelAlgorithmNonParametricModel ResourceMap key allows selecting the preferred one. Default value is HistogramOne each marginal is estimated, we use the Spearman independence test on each component pair to decide whether an independent copula. In case of non independence, we rely on a
NormalCopula
. Parameters
 sample
Sample
Input sample.
 sample
 Returns
 distribution
Distribution
Input distribution.
 distribution

getClassName
()¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getDistribution
()¶ Accessor to the joint probability density function of the physical input vector.
 Returns
 distribution
Distribution
Joint probability density function of the physical input vector.
 distribution

getId
()¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getName
()¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getNoise
()¶ Observation noise variance accessor.
 Parameters
 noisesequence of positive float
The noise variance of each output value.

getObjectiveFunction
()¶ Accessor to the loglikelihood function that writes as argument of the covariance’s model parameters.
Notes
The loglikelihood function may be useful for some postprocessing: maximization using external optimizers for example.
Examples
Create the model and the samples:
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x0'], ['x0 * sin(x0)']) >>> inputSample = ot.Sample([[1.0], [3.0], [5.0], [6.0], [7.0], [8.0]]) >>> outputSample = f(inputSample)
Create the algorithm:
>>> basis = ot.ConstantBasisFactory().build() >>> covarianceModel = ot.SquaredExponential(1) >>> algo = ot.GeneralLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis) >>> algo.run()
Get the loglikelihood function:
>>> likelihoodFunction = algo.getObjectiveFunction()

getOptimizationAlgorithm
()¶ Accessor to solver used to optimize the covariance model parameters.
 Returns
 algorithm
OptimizationAlgorithm
Solver used to optimize the covariance model parameters. Default optimizer is
TNC
 algorithm

getOptimizationBounds
()¶ Optimization bounds accessor.
 Returns
 bounds
Interval
Bounds for covariance model parameter optimization.
 bounds

getOptimizeParameters
()¶ Accessor to the covariance model parameters optimization flag.
 Returns
 optimizeParametersbool
Whether to optimize the covariance model parameters.

getResult
()¶ Get the results of the metamodel computation.
 Returns
 result
GeneralLinearModelResult
Structure containing all the results obtained after computation and created by the method
run()
.
 result

getShadowedId
()¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

getVisibility
()¶ Accessor to the object’s visibility state.
 Returns
 visiblebool
Visibility flag.

hasName
()¶ Test if the object is named.
 Returns
 hasNamebool
True if the name is not empty.

hasVisibleName
()¶ Test if the object has a distinguishable name.
 Returns
 hasVisibleNamebool
True if the name is not empty and not the default one.

run
()¶ Compute the response surface.
Notes
It computes the response surface and creates a
GeneralLinearModelResult
structure containing all the results.

setDistribution
(distribution)¶ Accessor to the joint probability density function of the physical input vector.
 Parameters
 distribution
Distribution
Joint probability density function of the physical input vector.
 distribution

setName
(name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setNoise
(noise)¶ Observation noise variance accessor.
 Parameters
 noisesequence of positive float
The noise variance of each output value.

setOptimizationAlgorithm
(solver)¶ Accessor to the solver used to optimize the covariance model parameters.
 Parameters
 algorithm
OptimizationAlgorithm
Solver used to optimize the covariance model parameters.
 algorithm

setOptimizationBounds
(optimizationBounds)¶ Optimization bounds accessor.
 Parameters
 bounds
Interval
Bounds for covariance model parameter optimization.
 bounds
Notes
Parameters involved by this method are:
Scale parameters,
Amplitude parameters if output dimension is greater than one or analytical sigma disabled,
Additional parameters.
Lower & upper bounds are defined in resource map. Default lower upper bounds value for all parameters is and defined thanks to the GeneralLinearModelAlgorithmDefaultOptimizationLowerBound resource map key.
For scale parameters, default upper bounds are set as times the difference between the max and min values of X for each coordinate, X being the (transformed) input sample. The value is defined in resource map (GeneralLinearModelAlgorithmDefaultOptimizationScaleFactor).
Finally for other parameters (amplitude,…), default upper bound is set to (corresponding resource map key is GeneralLinearModelAlgorithmDefaultOptimizationUpperBound)

setOptimizeParameters
(optimizeParameters)¶ Accessor to the covariance model parameters optimization flag.
 Parameters
 optimizeParametersbool
Whether to optimize the covariance model parameters.

setShadowedId
(id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setVisibility
(visible)¶ Accessor to the object’s visibility state.
 Parameters
 visiblebool
Visibility flag.